Improving many objective optimisation algorithms using objective dimensionality reduction

Abstract

Many-objective optimisation problems (MaOPs) have recently received a considerable attention from researchers. Due to the large number of objectives, MaOPs bring serious difficulties to existing multi-objective evolutionary algorithms (MOEAs). The major difficulties includes the poor scalability, the high computational cost and the difficulty in visualisation. A number of many-objective evolutionary algorithms (MaOEAs) has been proposed to tackle MaOPs, but existing MaOEAs have still faced with the difficulties when the number of objectives increases. Real-world MaOPs often have redundant objectives that are not only inessential to describe the Pareto-optimal front, but also deteriorate MaOEAs. A common approach to the problem is to use objective dimensionality reduction algorithms to eliminate redundant objectives. By removing redundant objectives, objective reduction algorithms can improve the search efficiency, reduce computational cost, and support for decision making. The performance of an objective dimensionality reduction strongly depends on nondominated solutions generated by MOEAs/MaOEAs. The impact of objective reduction algorithms on MOEAs and vice versa have been widely investigated. However, the impact of objective reduction algorithms on MaOEAs and vice versa have been rarely investigated. This paper studies the interdependence of objective reduction algorithms on MaOEAs. Experimental results show that combining an objective reduction algorithm with an MOEA can only successfully remove redundant objectives when the total number of objectives is small. In contrast, combining the objective reduction algorithm with an MaOEA can successfully remove redundant objectives even when the total number of objectives is large. Experimental results also show that objective reduction algorithms can significantly improve the performance of MaOEAs.

Appendices

Appendix

This section further investigates the proposed methods when using a clustering method for objective dimensionality reduction.

Integrating clustering objective dimensionality reduction algorithm into MaOEAs

Based on correlation coefficient (where $\rho (x,y)$ is the correlation coefficient between random variables x and y, the range of $\rho$ is from ${-}$ 1 to 1), Jaimes et al. [33] used $(1-\rho ) \in [0,2]$ to measure the degree of correlation between two objectives in approximation set of Pareto Front in MOPs. In which, zero value indicates that objectives x and y are completely positively correlated and a value of 2 indicates that x and y are completely negatively correlated. A negative correlation between two objectives means that one objective increases while the other decreases and vice versa. On the other hand, if the correlation is positive, then both objectives increase or decrease at the same time. Following this way, the more negative correlation between two objectives leads to the more conflict between the objectives. In [33], based on a correlation matrix of a non-dominated set obtained using an evolutionary algorithm, the objective set is first divided into homogeneous neighborhoods. The distance between the objectives is considered as the conflict between the objectives. Thereafter, the most compact neighborhood is chosen, and all the objectives in it except the center one are removed, as they are the least conflicting.

MICA-NORMOEA and OC-ORA algorithms are developed in [23, 24], respectively. In these algorithms, interdependence coefficient matrix is calculated, then PAM clustering algorithm [29] and NSGA-II [15] are invoked iteratively to reduce the redundant objectives until criterion is satisfied. The main different between these methods with LPCA is the relationship between pair of objectives. While LPCA use linear relationship, the method represents nonlinear one.

The framework of these algorithms (MICA-NORMOEA and OC-ORA) is shown

Step 1. Set an iteration counter $t = 0$; original objective set is $F_t={f_1,f_2,\ldots ,f_M }$, and the number of predefined clusters is k.

Step 2. Initialize a random population $P_t$ run NSGA-II corresponding to $F_t$ and obtain a non-dominated set $A_t$

Step 3. Calculate the interdependence coefficient matrix based on the non-dominated set $A_t$ and use the PAM clustering algorithm to divide the objective set $F_t$ into k clusters.

Step 4. According to the clusters of objective set $F_t$ obtained in Step 3, remove one of the redundant or the most interdependent objective from $F_t$ according to the above objective reduction rules, and the remaining objective set is denoted as $F_{t+1}$

Step 5. If $F_t$ = $F_{t+1}$ then stop; else $t:=t+1$; $F_t:= F_{t+1}$; return to Step 2.

The results

These are done same as in Sect. 3. When calculating interdependence, the number of subintervals is set as 20, and the threshold $\theta$ is set as 0.9.

Table 7 shows the mean and standard deviation (in parentheses) of GD and IGD of five MaOEAs including GrEA, KnEA, NSGAIII, RVEA*, and $\theta$-DEA. $IGD_{1}$ and $GD_{1}$ refer to IGD and GD of the MaOEAs without combining with objective dimensionality reduction algorithm, respectively. $IGD_{2}$ and $GD_{2}$ refer to IGD and GD of the MaOEAs combining with clustering objective dimensionality reduction (OCA-ORA) for removing redundant objectives, respectively. The table also shows the mean and standard deviation of the number of objectives which are retained after carrying out objective reduction. The table indicates the performance of the combination of MaOEAs and clustering objective reduction is significantly better than MaOEAs alone in almost all cases. In detail, $IGD_2$ is significant better than $IGD_1$ on 21 of 30 cases, and $GD_2$ is also significant better than $GD_1$ in 28 of 30 cases. Moreover, $IGD_2$ is only significant worse than $IGD_1$ on 4 of 30 cases, and $GD_2$ is significant worse than $GD_1$ in 2 of 30 cases.

In summary, the proposed methods can be combined with different objective dimensionality reduction methods to improve evolutionary computation many objective optimisation algorithms.

Table 7 The values of IGD, GD of true Pareto ($IGD_1$, $GD_1$); the number of objective retain (R) and IGD, GD ($IGD_2$, $GD_2$) after carrying out clustering objective dimensionality reduction